Theorem xadd4d 9668

Description: Rearrangement of 4 terms in a sum for extended addition, analogous to add4d 7931. (Contributed by Alexander van der Vekens, 21-Dec-2017.)

Hypotheses

Ref	Expression
xadd4d.1	|- ( ph -> ( A e. RR* /\ A =/= -oo ) )
xadd4d.2	|- ( ph -> ( B e. RR* /\ B =/= -oo ) )
xadd4d.3	|- ( ph -> ( C e. RR* /\ C =/= -oo ) )
xadd4d.4	|- ( ph -> ( D e. RR* /\ D =/= -oo ) )

Assertion

Ref	Expression
xadd4d	|- ( ph -> ( ( A +e B ) +e ( C +e D ) ) = ( ( A +e C ) +e ( B +e D ) ) )

Proof of Theorem xadd4d

Step	Hyp	Ref	Expression
1		xadd4d.3	. . . 4 |- ( ph -> ( C e. RR* /\ C =/= -oo ) )
2		xadd4d.2	. . . 4 |- ( ph -> ( B e. RR* /\ B =/= -oo ) )
3		xadd4d.4	. . . 4 |- ( ph -> ( D e. RR* /\ D =/= -oo ) )
4		xaddass 9652	. . . 4 |- ( ( ( C e. RR* /\ C =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( D e. RR* /\ D =/= -oo ) ) -> ( ( C +e B ) +e D ) = ( C +e ( B +e D ) ) )
5	1, 2, 3, 4	syl3anc 1216	. . 3 |- ( ph -> ( ( C +e B ) +e D ) = ( C +e ( B +e D ) ) )
6	5	oveq2d 5790	. 2 |- ( ph -> ( A +e ( ( C +e B ) +e D ) ) = ( A +e ( C +e ( B +e D ) ) ) )
7		xadd4d.1	. . . 4 |- ( ph -> ( A e. RR* /\ A =/= -oo ) )
8	1	simpld 111	. . . . 5 |- ( ph -> C e. RR* )
9	3	simpld 111	. . . . 5 |- ( ph -> D e. RR* )
10	8, 9	xaddcld 9667	. . . 4 |- ( ph -> ( C +e D ) e. RR* )
11		xaddnemnf 9640	. . . . 5 |- ( ( ( C e. RR* /\ C =/= -oo ) /\ ( D e. RR* /\ D =/= -oo ) ) -> ( C +e D ) =/= -oo )
12	1, 3, 11	syl2anc 408	. . . 4 |- ( ph -> ( C +e D ) =/= -oo )
13		xaddass 9652	. . . 4 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( ( C +e D ) e. RR* /\ ( C +e D ) =/= -oo ) ) -> ( ( A +e B ) +e ( C +e D ) ) = ( A +e ( B +e ( C +e D ) ) ) )
14	7, 2, 10, 12, 13	syl112anc 1220	. . 3 |- ( ph -> ( ( A +e B ) +e ( C +e D ) ) = ( A +e ( B +e ( C +e D ) ) ) )
15	2	simpld 111	. . . . . . 7 |- ( ph -> B e. RR* )
16		xaddcom 9644	. . . . . . 7 |- ( ( C e. RR* /\ B e. RR* ) -> ( C +e B ) = ( B +e C ) )
17	8, 15, 16	syl2anc 408	. . . . . 6 |- ( ph -> ( C +e B ) = ( B +e C ) )
18	17	oveq1d 5789	. . . . 5 |- ( ph -> ( ( C +e B ) +e D ) = ( ( B +e C ) +e D ) )
19		xaddass 9652	. . . . . 6 |- ( ( ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) /\ ( D e. RR* /\ D =/= -oo ) ) -> ( ( B +e C ) +e D ) = ( B +e ( C +e D ) ) )
20	2, 1, 3, 19	syl3anc 1216	. . . . 5 |- ( ph -> ( ( B +e C ) +e D ) = ( B +e ( C +e D ) ) )
21	18, 20	eqtr2d 2173	. . . 4 |- ( ph -> ( B +e ( C +e D ) ) = ( ( C +e B ) +e D ) )
22	21	oveq2d 5790	. . 3 |- ( ph -> ( A +e ( B +e ( C +e D ) ) ) = ( A +e ( ( C +e B ) +e D ) ) )
23	14, 22	eqtrd 2172	. 2 |- ( ph -> ( ( A +e B ) +e ( C +e D ) ) = ( A +e ( ( C +e B ) +e D ) ) )
24	15, 9	xaddcld 9667	. . 3 |- ( ph -> ( B +e D ) e. RR* )
25		xaddnemnf 9640	. . . 4 |- ( ( ( B e. RR* /\ B =/= -oo ) /\ ( D e. RR* /\ D =/= -oo ) ) -> ( B +e D ) =/= -oo )
26	2, 3, 25	syl2anc 408	. . 3 |- ( ph -> ( B +e D ) =/= -oo )
27		xaddass 9652	. . 3 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) /\ ( ( B +e D ) e. RR* /\ ( B +e D ) =/= -oo ) ) -> ( ( A +e C ) +e ( B +e D ) ) = ( A +e ( C +e ( B +e D ) ) ) )
28	7, 1, 24, 26, 27	syl112anc 1220	. 2 |- ( ph -> ( ( A +e C ) +e ( B +e D ) ) = ( A +e ( C +e ( B +e D ) ) ) )
29	6, 23, 28	3eqtr4d 2182	1 |- ( ph -> ( ( A +e B ) +e ( C +e D ) ) = ( ( A +e C ) +e ( B +e D ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   =/= wne 2308  (class class class)co 5774   -oocmnf 7798   RR*cxr 7799   +ecxad 9557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1re 7714  ax-addrcl 7717  ax-addcom 7720  ax-addass 7722  ax-rnegex 7729

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-pnf 7802  df-mnf 7803  df-xr 7804  df-xadd 9560

This theorem is referenced by:  xnn0add4d  9669